Jeśli występ się uda, nasza nauczycielka kupi dzieciom lody po drodze do domu.

Jeśli means if and introduces a condition. In this sentence, the condition is występ się uda — if the performance goes well / is successful.

You could also see jeżeli, which means the same thing and is a bit more formal or neutral in style. In everyday Polish, jeśli is very common.

Udać się is an idiomatic verb meaning to succeed, to turn out well, or to go well.

So:

• występ się uda = the performance will be successful / will go well

This is not the same as the ordinary verb udać by itself. You usually learn it as the set expression udać się.

Examples:

• Plan się udał. = The plan worked out.
• Egzamin się nie udał. = The exam did not go well.

Because udać się is a perfective verb.

In Polish, perfective verbs do not normally have a true present-tense meaning. Their present-looking forms are used to talk about the future.

So:

• uda się = it will succeed / it will go well
• kupi = she will buy

By contrast, the imperfective verb udawać się gives forms like udaje się, which can describe repeated or ongoing success, but that is not what this sentence is doing.

Here się does not literally mean itself. In many Polish verbs, się is just part of the verb expression.

So udać się should be learned as a whole unit. In this sentence:

• występ się uda = the performance will go well

It is better not to translate się word-for-word here.

Also, Polish often places się in the second position or near the verb, so you may also see different word orders in other sentences, depending on style and emphasis.

Because it is the subject of the main clause, so it is in the nominative case.

• nasza = feminine singular nominative of nasz (our)
• nauczycielka = feminine singular nominative (teacher, female)

Together:

• nasza nauczycielka = our teacher

If the case changed, the form of nasza and possibly the noun’s role in the sentence would change too.

Because kupić komuś coś means to buy someone something.

That pattern uses:

• the person receiving the thing in the dative
• the thing being bought in the accusative

So here:

• dzieciom = to the children / for the children → dative plural
• lody = ice cream → accusative plural

Compare:

• Kupię mamie kwiaty. = I’ll buy my mother flowers.
• Dał dzieciom prezent. = He gave the children a present.

Yes — lody means ice cream, but in Polish it is very often used as a plural noun.

So even when English uses singular ice cream, Polish commonly says lody.

Examples:

• Jemy lody. = We’re eating ice cream.
• Kupię lody. = I’ll buy ice cream.

Grammatically, lody is plural, even though the English meaning may feel singular.

Because kupić is the perfective verb, and here it means a single completed future action: she will buy.

• kupi = she will buy
• kupuje = she is buying / she buys

So in this sentence, kupi fits because the meaning is one future event that will happen if the condition is met.

Compare:

• kupuje lody = she is buying ice cream / she buys ice cream
• kupi lody = she will buy ice cream

Because the sentence begins with a subordinate if-clause:

• Jeśli występ się uda, ...

In Polish, when a subordinate clause comes before the main clause, it is separated by a comma.

So the structure is:

• Jeśli występ się uda, = subordinate clause
• nasza nauczycielka kupi dzieciom lody po drodze do domu. = main clause

This is similar to English punctuation in sentences like If the performance goes well, our teacher will buy the children ice cream...

It means on the way home.

Literally:

• po drodze = on the way
• do domu = to home / homeward

So the whole phrase means that the teacher will buy the ice cream while going back home with the children, not necessarily after arriving home.

Also note:

• do requires the genitive
• dom becomes domu after do

Yes. Polish word order is relatively flexible because cases show the grammatical roles.

So these are all possible, with slightly different emphasis:

• Nasza nauczycielka kupi dzieciom lody.
• Nasza nauczycielka kupi lody dzieciom.
• Dzieciom kupi lody nasza nauczycielka.

The original order sounds natural and neutral. Putting dzieciom before lody often highlights the recipients a bit: she’ll buy the children ice cream.

Sometimes, yes, but the exact meaning changes.

• występ usually means a performance, especially by performers, students, singers, dancers, etc.
• przedstawienie is more like a play or stage production
• show exists in Polish too, but it is a borrowing and has a different style and tone

So występ is a very natural choice if this is a school performance or presentation.

Yes, in meaning it is.

Both verbs refer to the future:

• występ się uda = the performance will go well
• nauczycielka kupi = the teacher will buy

But in Polish, with perfective verbs, this future is expressed by forms that look like present-tense forms.

So although uda and kupi may look short and simple, they are future in meaning here.